Abstract: In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes with values in a Banach space ; namely, is of class (LD), and the stochastic measure is bounded and strongly additive in (in particular, if is bounded in and ) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with -integrable semivariation or -integrable variation, the conditions of to be bounded and have bounded semivariation are superfluous. For processes with -integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.